Average Error: 29.9 → 0.5
Time: 6.2s
Precision: 64
\[\sqrt[3]{x + 1} - \sqrt[3]{x}\]
↓
\[\frac{1}{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1} + \left(\sqrt[3]{x} \cdot \sqrt[3]{x} + \sqrt[3]{x + 1} \cdot \sqrt[3]{x}\right)}\]
\sqrt[3]{x + 1} - \sqrt[3]{x}
↓
\frac{1}{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1} + \left(\sqrt[3]{x} \cdot \sqrt[3]{x} + \sqrt[3]{x + 1} \cdot \sqrt[3]{x}\right)}
double f(double x) {
        double r69719 = x;
        double r69720 = 1.0;
        double r69721 = r69719 + r69720;
        double r69722 = cbrt(r69721);
        double r69723 = cbrt(r69719);
        double r69724 = r69722 - r69723;
        return r69724;
}
↓
double f(double x) {
        double r69725 = 1.0;
        double r69726 = x;
        double r69727 = r69726 + r69725;
        double r69728 = cbrt(r69727);
        double r69729 = r69728 * r69728;
        double r69730 = cbrt(r69726);
        double r69731 = r69730 * r69730;
        double r69732 = r69728 * r69730;
        double r69733 = r69731 + r69732;
        double r69734 = r69729 + r69733;
        double r69735 = r69725 / r69734;
        return r69735;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 29.9

    \[\sqrt[3]{x + 1} - \sqrt[3]{x}\]
  2. Using strategy rm

  3. Applied add-exp-log29.9

    \[\leadsto \color{blue}{e^{\log \left(\sqrt[3]{x + 1} - \sqrt[3]{x}\right)}}\]
  4. Using strategy rm

  5. Applied flip3--29.9

    \[\leadsto e^{\log \color{blue}{\left(\frac{{\left(\sqrt[3]{x + 1}\right)}^{3} - {\left(\sqrt[3]{x}\right)}^{3}}{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1} + \left(\sqrt[3]{x} \cdot \sqrt[3]{x} + \sqrt[3]{x + 1} \cdot \sqrt[3]{x}\right)}\right)}}\]

  6. Applied log-div29.9

    \[\leadsto e^{\color{blue}{\log \left({\left(\sqrt[3]{x + 1}\right)}^{3} - {\left(\sqrt[3]{x}\right)}^{3}\right) - \log \left(\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1} + \left(\sqrt[3]{x} \cdot \sqrt[3]{x} + \sqrt[3]{x + 1} \cdot \sqrt[3]{x}\right)\right)}}\]

  7. Taylor expanded around 0 2.6

    \[\leadsto e^{\color{blue}{\log 1} - \log \left(\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1} + \left(\sqrt[3]{x} \cdot \sqrt[3]{x} + \sqrt[3]{x + 1} \cdot \sqrt[3]{x}\right)\right)}\]
  8. Using strategy rm

  9. Applied diff-log2.6

    \[\leadsto e^{\color{blue}{\log \left(\frac{1}{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1} + \left(\sqrt[3]{x} \cdot \sqrt[3]{x} + \sqrt[3]{x + 1} \cdot \sqrt[3]{x}\right)}\right)}}\]

  10. Applied rem-exp-log0.5

    \[\leadsto \color{blue}{\frac{1}{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1} + \left(\sqrt[3]{x} \cdot \sqrt[3]{x} + \sqrt[3]{x + 1} \cdot \sqrt[3]{x}\right)}}\]

  11. Final simplification0.5

    \[\leadsto \frac{1}{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1} + \left(\sqrt[3]{x} \cdot \sqrt[3]{x} + \sqrt[3]{x + 1} \cdot \sqrt[3]{x}\right)}\]

Reproduce

herbie shell --seed 2020024 
(FPCore (x)
  :name "2cbrt (problem 3.3.4)"
  :precision binary64
  (- (cbrt (+ x 1)) (cbrt x)))